A303048 Number of total dominating sets in the n-triangular (Johnson) graph.

0, 4, 54, 918, 31232, 2059624, 266734812, 68574627036, 35160753222400, 36021330363615408, 73782362964470935112, 302225854825997535378632, 2475866675779140063716682240, 40564755806137338166417907530592, 1329227401912999475682655581004557840

Offset: 2

Author

Eric W. Weisstein, Apr 17 2018

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 2..50
• Eric Weisstein's World of Mathematics, Johnson Graph
• Eric Weisstein's World of Mathematics, Total Dominating Set
• Eric Weisstein's World of Mathematics, Triangular Graph

Crossrefs

Cf. A006129, A290847, A298104, A303227, A304561.

Programs

Mathematica

b[n_] := Sum[(-1)^(n - k) Binomial[n, k] 2^Binomial[k, 2], {k, 0, n}]
Table[Sum[(-1)^k Binomial[n, 2 k] (2 k)!/(2^k k!) (b[n - 2 k] + (n - 2 k) b[n - 2 k - 1]), {k, 0, Floor[n/2]}], {n, 2, 20}]

PARI

\\ here b(n) is A006129
b(n)=sum(k=0, n, (-1)^(n-k)*binomial(n, k)*2^binomial(k, 2));
a(n)=sum(k=0, n\2, (-1)^k*binomial(n,2*k)*(2*k)!/(2^k*k!)*(b(n-2*k) + (n-2*k)*b(n-2*k-1))); \\ Andrew Howroyd, Apr 20 2018

Formula

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n,2*k)*(2*k-1)!!*A290847(n-2*k). - Andrew Howroyd, Apr 20 2018

Extensions

a(8)-a(16) from Andrew Howroyd, Apr 20 2018